Asymptotically Most Powerful Rank-Order Tests

Having observed Xᵢ = + cᵢ + Yᵢ, we test the hypothesis = 0 against the alternative > 0. We suppose that the square root of the probability density f (x) of the residuals Yᵢ possesses a quadratically integrable derivative and define a class of rank order tests, which are asymptotically most powerful for given f. The main result is exposed in the following succession: theorem, corollaries and examples, comments, preliminaries and proof. The proof is based on results by Hajek 6 and LeCam 8, 9. Section 6 deals with asymptotic efficiency of rank-order tests, which is shown, on the basis of Mikulski's results 10, to be presumably never less than the asymptotic efficiency of corresponding parametric tests of Neyman's type 11. This would extend the well-known result obtained by Chernoff and Savage 2 for the Student t-test. Furthermore, it is shown that the efficiency may be negative, i. e. , asymptotic power may be less than the asymptotic size. In Section 7 we consider parallel rank-order tests of symmetry for judging paired comparisons. Section 8 is devoted to rank-order tests for densities such that (f (x) ) ^1{2} does not possess a quadratically integrable derivative. In Section 9, we construct a test which is asymptotically most powerful simultaneously for all densities f (x) such that (f (x) ) ^1{2} possesses a quadratically integrable derivative.